Summary

General Principles

The Nucleus

The nucleus is a small, dense, positive core at the center of an atom.

The figure illustrates the protons and neutrons held together in a nucleus.

$Z$ protons: charge $+ e$ , spin $\frac{1}{2}$

$N$ neutrons: charge 0, spin $\frac{1}{2}$

The mass number is $A = Z + N$ .

The nuclear radius is $R = r_{0} A^{1 / 3}$ , where $r_{0} = 1.2$ fm. Typical radii are a few fm.

Nuclear forces

Attractive strong force	Repulsive electric force
Acts between any two nucleons Is short range, $< 3 fm$ Is felt between nearest neighbors	Acts between two protons Is long range Is felt across the nucleus

Nuclear Stability

Most nuclei are not stable. Unstable nuclei undergo radioactive decay. Stable nuclei cluster along the line of stability in a plot of the isotopes.

The figure illustrates the line of stability in an N versus Z graph.

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Three mechanisms by which unstable nuclei decay:

Decay	Particle	Mechanism	Energy	Penetration
$α$	$^{4} He$ nucleus	tunneling	few MeV	low
$β$	$e^{-}$	$n \to p^{+} + e^{-}$	$\approx 1 MeV$	medium
	$e^{+}$	$p^{+} \to n + e^{+}$	$\approx 1 MeV$	medium
$γ$	photon	quantum jump	$\approx 1 MeV$	high

Important Concepts

Shell model

Each nucleon moves with an average potential energy due to all other nucleons.

The figure illustrates the neutron and proton potential-energy wells of a high Z nucleus.

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Curve of binding energy

The average binding energy per nucleon has a broad maximum at $A \approx 60$ .

The figure illustrates the binding energy curve.

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Applications

Radioactive decay

The number of undecayed nuclei decreases exponentially with time $t$ :

\begin{array}{l} N & = & N_{0} exp (- t / τ) \\ = & N_{0} {(1 / 2)}^{t / t_{1 / 2}} \end{array}

The lifetime $τ$ is $1 / r$ , where $r$ is the decay rate.

The half-life

t_{1 / 2} = τ_{} ln 2 = 0.693 τ

is the time in which half of any sample decays.

The figure illustrates the radioactive decay.

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Measuring radiation

The activity $R = N / τ$ of a radioactive sample, measured in becquerels or curies, is the number of decays per second.

The absorbed dose is measured in grays, where

1 Gy \equiv 1.00 J/kg of absorbed energy

The relative biological effectiveness (RBE) is the biological effect of a dose relative to the biological effects of x rays.

The dose equivalent is measured in Sv, where $Sv = Gy \times RBE$ . One Sv of radiation produces the same biological effect regardless of the type of radiation. Dose equivalent is also measured in rem, where $1 rem = 0.010 Sv$ .